A study of pushdown games

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A Study of Pushdown Games

Von der Fakultät für Mathematik, Informatik und Naturwissenschaf-ten der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation

vorgelegt von

Dipl.-Inform.

Wladimir Fridman

aus Sankt Petersburg

Berichter:

Universitätsprofessor Dr. Dr.h.c. Wolfgang Thomas Lecturer Dr. Sven Schewe

Tag der mündlichen Prüfung: 29. Januar 2013

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar.

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Zusammenfassung

Unendliche Zwei-Personen-Spiele sind von wesentlicher Bedeutung in der Informatik, denn sie stellen einen algorithmischen Rahmen für die Unter-suchung von nicht-terminierenden reaktiven Systemen bereit. Üblicherweise werden unendlichen Spiele durch eine ω-Sprache spezifiziert, die alle gewin-nenden Partien für einen der beiden Spieler enthält, oder durch einen Spiel-graphen und eine Gewinnbedingung auf den unendlichen Pfaden durch diesen Graphen. Viele algorithmische Resultate sind bekannt für den Fall von regu-lären Spezifikationen und für endliche Spielgraphen mit Gewinnbedingungen wie der Mull oder Paritätsbedingung, die reguläre Problemstellungen er-fassen. Die Ergebnisse der vorliegenden Arbeit behandeln eine Klasse von nicht-regulären Spezifikationen und eine Klasse von unendlichen Spielgra-phen, nämlich solche, die durch Pushdown-Automaten spezifiziert werden, d.h. wir betrachten kontextfreie Spezifikationen und Pushdown-Spielgraphen mit Muller- oder Paritätsgewinnbedingungen. Wir erweitern zahlreiche zen-trale für reguläre Spiele bekannte Resultate auf die Klasse der Pushdown-Spiele. Dabei konzentrieren wir uns auf folgende vier Problemstellungen.

Zunächst analysieren wir, wie das Format einer Gewinnstrategie mit dem Typ der Spezifikation zusammenhängt. Dabei stellen wir für etliche kontext-freie Fälle eine Verbindung zwischen den Formaten der Spezifikationen und den Formaten der entsprechenden Gewinnstrategien her, zeigen aber auch Fälle von kontextfreien Spielen auf, wo diese Übereinstimmung nicht gilt.

Zweitens untersuchen wir Delay-Spiele mit kontextfreien Gewinnbedin-gungen. In einem Delay-Spiel hat einer der beiden Spieler die Möglichkeit, seine Züge für eine gewisse Zeit hinauszuzögern, damit wird eine Voraus-schau auf die Züge des Gegners erzielt. Wir klären, ob der Gewinner eines deterministisch kontextfreien Delay-Spiels berechnet werden kann, und wie groß die erforderliche Vorausschau zum Gewinnen solcher Spiele ist.

Drittens untersuchen wir das Synthese-Problem für verteilte Systeme, welche aus mehreren kooperierenden Komponenten bestehen, die miteinan-der und mit miteinan-der Umgebung kommunizieren. Dabei wird die Kommunikati-onsstruktur durch eine Architektur spezifiziert. Es werden beide Hauptkon-zepte studiert, das der globalen und das der lokalen Spezifikationen. Wir ge-ben eine Charakterisierung der entscheidbaren Architekturen für lokale Spe-zifikationen an, welche regulär oder deterministisch kontextfrei sein können. Außerdem zeigen wir Unentscheidbarkeit des verteilten Synthese-Problems für globale deterministisch kontextfreie Spezifikationen.

Schließlich wird das Problem behandelt, ob der Gewinner eines unendli-chen Spiels bereits nach einem endliunendli-chen Partiepräfix bestimmt werden kann. Aufbauend auf den Resultaten für den Fall der endlichen Spielgraphen füh-ren wir eine Paritätsspiele endlicher Dauer auf Pushdown-Graphen ein und

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Graphen. Dies ergibt eine neue Reduktionsmethode, mit welcher der Gewin-ner eines Pushdown-Spiels durch Lösen eines Erreichbarkeitsspiels auf einem endlichen Spielgraphen bestimmt werden kann.

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Abstract

Infinite two-player games are of interest in computer science since they pro-vide an algorithmic framework for the study of reactive nonterminating sys-tems. Usually, an infinite game is specified by an ω-language containing all winning plays for one of the two players or by a game graph and a winning condition on infinite paths through this graph. Many algorithmic results are known for the case of regular specifications and for finite game graphs with winning conditions like parity or Muller conditions capturing regular objec-tives. The present thesis offers results that also cover a class of nonregular specifications as well as a class of infinite game graphs, namely those spec-ified by pushdown automata, i.e, we consider contextfree specifications and pushdown game graphs with parity or Muller winning conditions. We ex-tend various central results known for regular games to the class of pushdown games. We focus on the following four questions.

Firstly, we analyze how the format of a pushdown winning strategy matches the type of the pushdown game specification. Here, we establish a strong connection between the formats of specifications and formats of corresponding winning strategies for several types of contextfree games, but we also exhibit cases of contextfree games where this correspondence fails.

Secondly, we investigate delay games with contextfree winning condi-tions. In such a game one of the players has the possibility to postpone his moves for some time, thus obtaining a lookahead on the moves of the op-ponent. We clarify whether the winner of a deterministic contextfree delay game can be determined effectively as well as what amount of lookahead is necessary to win such games.

Thirdly, we investigate the synthesis problem for distributed systems which consist of several cooperating components communicating with each other and with the environment. A distributed system is specified by an ar-chitecture comprising the communication structure of the system. Here, we study both main concepts, that of global and that of local specifications. We offer a complete characterization of the decidable architectures for local spec-ifications, which may be deterministic contextfree or regular. Moreover, we prove that, for global deterministic contextfree specifications, the distributed synthesis problem is undecidable.

Finally, we address the problem whether the winner of an infinite game can be already determined after a finite play prefix. Extending results for the case of finite game graphs, we introduce finite-duration parity pushdown games and establish their completeness for solving parity pushdown games. This yields a new reduction method to determine the winner of a pushdown game by solving a reachability game on a finite game graph.

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List of Figures

2.1 Finite Borel hierarchy . . . 21

3.1 Classes of visibly and non-visibly pushdown languages . . . . 33

3.2 Parity-StDVPA recognizing L2 ∈ VPLω\ DCFLω . . . 34

3.3 Parity-DV1CA recognizing L . . . 50

3.4 Parity blind one-counter game . . . 52

4.1 Part of a play encoding three configurations . . . 63

4.2 A prefix containing three blocks w0, w1 and w2 . . . 69

4.3 A play prefix containing violations of the successor relation 7− indicated by the players . . . 71

5.1 Induced graph GA of an architecture A . . . 82

5.2 Hierarchical architectures . . . 84

5.3 Undecidable architecture for global regular specifications . . . 86

5.4 Decidable architectures for global specifications from DCFLω . 88 5.5 Two-flanked pipeline with two system processes . . . 105

5.6 Undecidable architectures for local regular specifications . . . 113

5.7 Generic decidable architecture . . . 116

6.1 The pushdown game graph induced by S . . . 128

6.2 A finite path w, its stair positions and values of the stair-scoring functions . . . 129

6.3 The pushdown game graph induced by P . . . 131

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Chapter 1

Introduction

For several decades, the theory of finite automata has proved to be a powerful framework for the development of effective methods for program verification and program synthesis. A very active branch of research is concerned with non-terminating reactive systems, i.e., systems which continuously interact with an environment in contrast to programs that transform data and then terminate. Such non-terminating reactive systems occur in numerous con-texts, among them operating systems, control systems, and protocol design. A versatile conceptual model of non-terminating reactive systems is the no-tion of infinite games, played between an antagonistic environment and one or several system controllers, where plays correspond to infinite sequences of actions performed by the players.

The fundamental problem of this theory is Church’s Problem [Chu57,

Chu63], first proposed in the context of controller synthesis and then trans-formed to a game theoretical question by McNaughton [McN65]. It asks, for a given regular specification consisting of all correct behaviors of the consid-ered system, to decide whether there is a finite-state controller such that all possible behaviors of the system satisfy the given specification, and further-more to synthesize such a controller if it exists. In the framework of infinite games, Church’s Problem is formulated as a slightly modified version of a Gale-Stewart game [GS53]. So, it is to decide, for a winning condition given in terms of a logical formula, who is the winner of the game, and to provide a winning strategy for the winning player.

The first solution of this problem was offered by the pioneering work of Büchi and Landweber [BL69] who showed that, for specifications formulated in monadic second-order logic of one successor (over the natural numbers), the controller synthesis problem is decidable and finite-state solutions can be constructed effectively. Hence, for regular winning conditions, the winner can be decided and winning strategies can be implemented by finite automata with output. Starting from the Büchi-Landweber result, the research area

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of infinite games has undergone a very active and intensive development, in particular the study of the synthesis problem, due to the prospect of being able to construct system controllers automatically from the specifications, rather than to implement and then to verify already built systems.

Many variants and extensions of the basic setting have been explored in a multitude of different directions. For example, various specification formalisms were considered like linear-time temporal logic [MW81, PR89], branching-time temporal logic [EC82,KV97] or the modal µ-calculus [Koz83,

KV00b]. Or the case of incomplete information was considered [KV97] where systems have to satisfy specifications which also refer to signals not known to the controller, in contrary to the basic setting, where both agents have the entire knowledge about all actions performed so far. Another exten-sion is to provide the controller with a buffer allowing to store some actions performed by the environment, in this way the controller gains the possi-bility to postpone his actions for some time, which leads to so-called delay games [HL72, HKT10]. A further natural extension where both scenarios, incomplete information as well as delay, are conceivable is the extension to distributed systems [PR90,KV01,MT01,FS05] which consist of several com-ponents which communicate with each other and with the environment via certain communication channels. Hence, the task of the distributed synthe-sis is to construct several controllers, one for each process, such that every overall system behavior satisfies a given specification. In the framework of in-finite games, distributed synthesis corresponds to multi-player games where several cooperating controller players play against the antagonistic environ-ment player [MW03]. The task is then to find a set of winning strategies, one for each controller player, comprising a joint winning strategy for the sys-tem. Of course, there are much more extensions like, for instance, stochastic versions of reactive systems [CJH04,Zie04].

A prerequisite for Büchi-Landweber’s solution was the transformation of a logically defined game into a graph game, played on a finite game graph, where the winning condition is just concerned with the requirement that cer-tain vertices are visited infinitely often and others only finitely often. These games are known as Muller games over finite graphs. A classical approach for solving Muller games is the reduction to so-called parity games [Zie98] where every vertex of the game graph is assigned a natural number and the winning condition now is concerned with the smallest color visited infinitely often. A central topic is the extension of this game models to cover infinite game graphs which emerge when the specification formalisms for the synthe-sis problem go beyond regular languages. The essential case of games played on infinite graphs which has been shown to be decidable is the class of parity games played on pushdown graphs [Wal96]. These are configuration graphs of pushdown systems which, roughly speaking, are infinite state systems

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1.1 Contribution

obtained from finite state systems by augmenting them with an infinite last-in-first-out data structure, a pushdown stack. Pushdown systems naturally occur in the context of program analysis and compiler construction. More-over, the importance of this model in formal languages also arises from the equivalence between the corresponding language acceptors, pushdown au-tomata, and contextfree grammars, which yields definability of contextfree languages in terms of pushdown automata.

The technique for solving parity games played on pushdown graphs pro-posed by Walukiewicz [Wal96] is a reduction to a parity game played on a finite game graph. The core idea is to encode the information stored on the stack by some finite memory structure. For this, a sophisticated finite game graph is defined to simulate the pushdown game which intends one player to make predictions about the future of the play and the opponent checks correctness of this predictions. Another method is due to Kupferman and Vardi [KV00a], it uses an infinite tree that represents all possible stack contents and alternating two-way tree automata operating on this tree to simulate parity pushdown games. These basic methods will be presented in detail and adapted appropriately in chapters 3 and 6 to prove the main results of those chapters.

1.1 Contribution

This thesis investigates several refinements and extensions, which have been considered within the domain of games over finite game graphs, in the context of pushdown games. So, we study the possibility of lifting the central results known for regular games to the class of contextfree games. Let us sketch the particular contributions informally.

Connecting Game Specifications to Winning Strategies

The Büchi-Elgot-Trakhtenbrot result [Büc60,Elg61,Tra61] establishes a con-nection between monadic second-order logic and finite automata by show-ing that both formalisms are expressively equivalent and providshow-ing effective translations from one formalism to another and vice versa. By this corre-spondence, with the focus on the formats of game specifications and their solutions, the Büchi-Landweber theorem states that solutions of games with monadic second-order logic specifications are again definable by monadic sec-ond order logic. This result has been refined in several works, where a close conceptual connection between the types of the winning conditions on one hand and winning strategies on the other hand has been established. Ra-binovich and Thomas exhibited several fragments L of the monadic second-order logic, among them first second-order logic over (N, <) and its extension by

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modular counting quantifiers, first order logic over (N, S) with successor re-lation S and the quantifier-free first-order logic over (N, 0, +1), called strictly bounded logic, for which the mentioned correspondence holds [RT07]. I.e., games given by specifications definable in L are determined with winning strategies which are again definable in L. Selivanov showed an analogous re-sult for a further regular subclass stating that every game defined by an aperi-odic regular winning condition is determined by winning strategies which can be realized by aperiodic transducers [Sel07, Sel08]. Chaturvedi et al. study this correspondence in the recent paper [COT11] for subclasses of star-free languages by considering piecewise testable languages and languages from the dot-depth hierarchy [CB71].

We address this question relating game representations and correspond-ing solutions for several classes of contextfree games. Thereby, we consider Gale-Stewart games with various types of contextfree specifications emerg-ing from the synthesis problem, as well as more general games played on pushdown game graphs defined by several types of pushdown automata. We prove the existence of a strong connection between the formats of specifi-cations and formats of the corresponding winning strategies for plenty of cases. Among them, games played on configuration graphs of realtime push-down systems, visibly pushpush-down systems, and one-counter systems with par-ity winning conditions, as well as with stair parpar-ity winning conditions and Gale-Stewart games defined by corresponding pushdown automata. Further-more, we exhibit some cases for which this correspondence between games and solutions does not hold, namely for visibly and blind one-counter.

The results are partially based on the publication [Fri10].

Delay Games

The generalization of the game model where the strict alternation between the moves of the environment and the controller is modified by allowing the controller to postpone his moves for some time has been considered by Hosch and Landweber in [HL72]. This so called delay games capture systems provided with first-in-first-out buffers to store inputs coming from the environment. Using these buffers, the controller obtains a lookahead on the moves of the environment. Hosch and Landweber showed that it is decidable, whether a regular game can be won by the controller with bounded lookahead, i.e., using a finite buffer. This result was improved by Holtmann et al. [HKT10], who showed that if a regular game is won with arbitrary lookahead then it is also won with a bounded lookahead, which is at most doubly-exponential in the size of the parity automaton recognizing the winning condition.

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1.1 Contribution

stated in [HKT10] whether the winner of a deterministic contextfree delay game can be determined effectively and what amount of lookahead is nec-essary to win such games. We prove that, for fixed bounded lookahead, determining the winner is decidable. However, the question whether a given player can win a deterministic contextfree delay game with arbitrary looka-head is undecidable. Moreover, we characterize classes of delay functions, for which the problem of determining the winner is decidable, if the lookahead is restricted to the given class of delay functions. Furthermore, we establish non-elementary lower bounds on delay functions by showing that there ex-ist determinex-istic contextfree delay games such that the controller wins with some lookahead, however, if the lookahead is bounded by an elementary function, i.e., a k-fold exponential for some fixed natural number k, then the environment wins.

The results are partially based on joint work with Christof Löding and Martin Zimmermann published in [FLZ11].

Distributed Synthesis

Distributed systems are specified by architectures consisting of a set of coop-erating processes and a set of channels via which the processes communicate with each other and with the environment. Based on the work of Peterson and Reif on multi-player games [PR79], Pnueli and Rosner proved that in general distributed synthesis is undecidable for specifications in linear time temporal logic, however, for a special class of acyclic architectures, called pipelines, where the processes are linearly ordered and the information flows from the environment in direction of the worst informed process, they proved decidability for linear time temporal logic specifications [PR90]. Kupferman and Vardi extended this decidability result to some further classes of archi-tectures which also may contain cycles and to specifications in branching time temporal logic [KV01]. In [FS05], Finkbeiner and Schewe gave a full characterization of decidable architectures for specifications given in linear time, branching time temporal logic as well as in µ-calculus. Moreover, a uniform tree automata based synthesis algorithm for decidable architectures was provided. Madhusudan and Thiagarajan introduced the concept of lo-cal specifications [MT01], where the system specification is given by a set of local specifications, one for each system process. For regular local specifica-tions and the class of acyclic architectures, they gave a characterization of all decidable architectures.

We investigate the distributed synthesis problem for contextfree global specifications as well as local specifications which are regular or contextfree. We give a complete characterization of decidable architectures for the case of local specifications which extends the result of [MT01] in two structural

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as-pects. First, we consider general architectures where also cycles are allowed, and second, the local specifications may also be deterministic contextfree. We give an effective criterion which concerns the graph structure of the given architecture and the assignment of regular and contextfree local speci-fications to the individual processes. Moreover, for deterministic contextfree global specifications, we prove undecidability for almost all architectures. Only the corner cases corresponding to the nondistributed setting or the case where the environment does not send information, are decidable.

The results were obtained in collaboration with Bernd Puchala and are partially based on the publication [FP11].

Finite-Time Games

McNaughton introduced a finite-duration variant for Muller games played on finite game graphs using scoring functions which rate finite play prefixes of infinite plays [McN00]. The scoring functions1 _{give a condition on}

ter-mination of plays in a finite-time game, i.e., a play is stopped when some scoring function reaches a given threshold score value. McNaughton proved equivalence of Muller games and corresponding finite-time Muller games with factorial threshold score, i.e., both variants have the same winner. Thus, the winner of a Muller game on a finite game graph can be determined by solving a reachability game over a game graph, which is doubly-exponential in the size of the game graph of the original Muller game. This result was improved by Fearnley and Zimmermann, who showed that the constant threshold score value of three suffices for the equivalence of the corresponding games [FZ10]. Furthermore, a score-based reduction from Muller games to safety games evolved from this result which yields non-deterministic winning strategies called permissive strategies [Zim12,NRZ12]. This extends the work of Ber-net et al. [BJW02] on permissive strategies for parity games to Muller games. We introduce a new finite-duration variant for parity pushdown games which is required since the known results on finite game graphs don’t hold for infinite game graphs. We define stair-scoring functions by exploiting the in-trinsic structure of the pushdown graphs and prove the equivalence between parity pushdown games and corresponding finite-time versions with thresh-old stair-score which is exponential in the size of the underlying pushdown system. Moreover, we give an almost matching lower bound on the thresh-old stair-score such that the equivalence between the corresponding games holds, which is exponential in the cube root of the size of the underlying pushdown system. Our result yields a new reduction method to determine the winner of a pushdown game by solving a reachability game over a finite

1_{The idea of declaring the winner of a play after a finite number of rounds using scores}

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1.2 Organization of the thesis

game graph.

The results were obtained in collaboration with Martin Zimmermann and are partially based on [FZ12].

1.2 Organization of the thesis

The thesis is structured as follows. In the subsequent Chapter2we introduce the basic notions like finite automata, pushdown automata, tree automata, infinite games, strategies and game reductions, and fix our notations used throughout the thesis.

In Chapter3we present our results on the connection between contextfree specifications and pushdown winning strategies. The considered classes of contextfree languages are introduced in Section3.1. We state our main result in Section 3.2. First, the technique of Kupferman and Vardi is recalled in Subsection3.2.1. Subsequently, we prove our main result in Subsection3.2.2. Chapter 4 presents our results on contextfree delay games. After delay games are introduced formally in Section4.1, we prove our decidability and undecidability results and give a general criterion characterizing the sets of delay functions where decidability is guaranteed in Section4.2. Chapter4is concluded by presenting a lower bound for the lookahead in Section 4.3.

We handle distributed synthesis in Chapter 5. First, we fix our nota-tions and give some defininota-tions in Section 5.1 which are used throughout this chapter. Architectures are introduced in Section 5.2. We prove unde-cidability for architectures with global deterministic contextfree specifica-tions in Section 5.3. The following Section 5.4 is composed of decidability and undecidability results for special cases of architectures with local spec-ifications, presented in Subsection 5.4.1 and Subsection 5.4.2, respectively. These results are assembled in the subsequent Section5.5where a complete characterization of the decidable architectures with regular or deterministic contextfree local specifications is presented.

Chapter 6 is concerned with finite-time games which we introduce for-mally in Section 6.1. There, we also define the new notion of stair-scoring functions for finite-time games over pushdown game graphs. In Section 6.2, Walukiewicz’s construction is recalled and adapted as it is needed in the fol-lowing Section6.3where we prove the equivalence between parity pushdown games and their corresponding finite-duration variants. Section6.4presents a lower bound on the threshold stair-score value for which the equivalence between parity pushdown games and their finite-duration variants holds.

Finally, we give a conclusion in Chapter7where we point out some open questions for further research.

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Acknowledgements

First, I would like to express my deep gratitude to my advisor Wolfgang Thomas who initiated this doctoral project and gave me the opportunity to work on the present topic. I am grateful for his constant support, valuable advice and professional guidance he gave me throughout the last three years. For me it was a great stroke of luck and personal enrichment to have him as my advisor as his versatile personality sets himself apart from the computer science community.

I would like to thank Sven Schewe for his kind and prompt readiness to act as an external reviewer as well as for his cordial and complaisant support in the last months before the examination. Moreover, his excellent PhD thesis was one of the initial works I was suggested to study when I started my research on distributed synthesis. Furthermore, I am grateful to Joost-Pieter Katoen and Thomas Seidl for joining my thesis committee.

It was a pleasure to work with Christof Löding, Bernd Puchala, and Martin Zimmermann, I wish to thank them for our fruitful collaborations.

I would like to thank Martin Lang, Daniel Neider, Roman Rabinovich, and Nora Schaal for proofreading parts of this thesis.

Furthermore, I am also indebted to a couple of my colleagues from the Chair Logic and Theory of Discrete Systems, the Research Group on Mathe-matical Foundations of Computer Science and from the Chair of Communi-cation and Distributed Systems who provided some diversion from the effort of doing research and from daily routine, especially Christof Löding for the juggling and passing breaks, Jörg Olschewski for innumerable discussions about anything and everything as well as for the short trips we undertook. Furthermore, I would also like to mention Helen Bolke-Hermanns, Namit Chaturvedi, Silke Cormann, Alex Kwiecien, and Alex Spelten.

Finally, I wish to thank my parents, my sister Emma, and my brother Alex for their support and encouragement, as well as my grandparents to whom I wish to dedicate this thesis. I want to offer my special thanks to my wife Nona for her support and patience she gave me in the last years.

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Chapter 2

Preliminaries

In this chapter we give the definitions and fix our notations which will be used throughout this work. After introducing some basic definitions in Sec-tion 2.1 we present the automata models of finite automata and pushdown automata over finite and infinte words as well as various kinds of tree au-tomata operating on infinite trees in Section 2.2. Section 2.3 is concerned with turn-based infinite two-player games. Gale-Stewart games, games on graphs as well as notions of winning conditions, winning strategies and game reductions are introduced.

2.1 Basic Definitions

The set of non-negative integers is denoted by N. For n ∈ N, let Par(n) = 0 if n is even, and Par(n) = 1 if n is odd. Furthermore, we denote the set of the first n elements of N by [n] = {0, . . . , n − 1}. For a set X, the power set of X is denoted by P(X) and the cardinally of X is denoted by |X|.

A finite nonempty set Σ of symbols or letters is called alphabet. For an alphabet Σ, the set of finite words is denoted by Σ∗. The length of a word w ∈ Σ∗ is denoted by |w|, and ε denotes the empty word, i.e., the word of length |ε| = 0. For a word w ∈ Σ∗ and a letter a ∈ Σ, the number of occurrences of a in w is denoted by |w|_a. We denote the set of words of length at most n by Σ≤n_{, for n ∈ N, and for the set of nonempty words} Σ∗\ {ε} we also write Σ+_{. The set of infinite words, also called ω-words, is}

denoted by Σω.

For a word w ∈ Σ∗∪ Σω _{and n ∈ N we write w(n) for the n-th letter of w}

where the first letter of w is w(0). If w is finite, we denote its last letter by last(w). The prefix of length n of a word w ∈ Σ∗∪Σω_{is denoted by pref}

n(w),

i.e., pref₀(w) = ε and pref_n(w) = w(0) · · · w(n − 1), for 0 < n ≤ |w|. For two words w ∈ Σ∗ and w0 ∈ Σ∗_{∪ Σ}ω_{, we write w v w}0 _{if w is a prefix of w}0_,

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a word w ∈ Σ∗ the reverse of w is denoted by rev(w), i.e., rev(ε) = ε and rev(ua) = a · rev(u), for u ∈ Σ∗ and a ∈ Σ.

A language over an alphabet Σ is a subset of Σ∗ or a subset of Σω. If it is not clear from the context whether a language contains finite or infinite words, we will also refer to languages containing finite word as ∗-languages and to those containing infinite words as ω-languages.

For a word w ∈ Σ∗∪ Σω_{, we define the occurrence set of w by}

Occ(w) = {a ∈ Σ | w(n) = a for some n ∈ N} , and the infinity set of w by

Inf(w) = {a ∈ Σ | w(n) = a for infinitely many n ∈ N} .

Let Σ0, . . . , Σn−1 be alphabets. For a cartesian product Σ =Q_i∈[n]Σi =

Σ0× · · · × Σn−1let dim(Σ) = n. For the cartesian product Σ =Qi∈[n]Σiand

a set X ⊆ [n] of indices, let the cartesian product restricted to alphabets with an index from X be denoted by ΣX = Q_i∈XΣi (here again, an ascending

order of indices is assumed).

The projection operator Pr_X: Σ → ΣX is defined by PrX(a) = (ai)i∈X,

for a = (a0, . . . , an−1) ∈ Σ. If X = {i} is a singleton set we will also write

Pri instead of Pr{i}. Moreover, we will also write PrΣX instead of PrX if we

do not refer to an explicit ordering of the components of Σ. We extend this definition in a natural way to words and languages. For a word w ∈ Σ∗∪ Σω_,

define PrX(w) = PrX(w(0))PrX(w(1)) · · · , and for a language L ⊆ Σ∗ or

L ⊆ Σω, define PrX(L) = {PrX(w) | w ∈ L}.

Moreover, for two ω-words w ∈ Σω and w0∈ (Σ0₎ω _{over the alphabets Σ}

and Σ0, by w_w0 ∈ (Σ×Σ0)ω_{we denote the ω-word with (w}__w0_{)(i) =} w(i) w0_(i),

for all i ∈ N.

2.2 Automata

2.2.1 Finite Automata

A (nondeterministic) finite automaton (NFA) A = (Q, Σ, Q_in, ∆, F ) consists of a finite set of states Q with the set of initial states Qin ⊆ Q, an input

alphabet Σ, a set of final states F ⊆ Q and a transition relation ∆ ⊆ Q × Σ × Q. An NFA is deterministic (DFA) if |Qin| = 1 and for all q ∈ Q

and all a ∈ Σ, |{q0 ∈ Q | (q, a, q0_{) ∈ ∆}| ≤ 1 is satisfied. In this case, we use}

a (partial) function δ : Q × Σ → Q to denote the transition relation ∆ and we just write qinto denote the set Qin= {qin}.

A run ρ of A on a word w ∈ Σ∗ is a finite sequence of states ρ = ρ(0) · · · ρ(|w|) such that ρ(0) ∈ Qin, and for every 0 ≤ i < |w|, we have

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2.2 Automata

(ρ(i), w(i), ρ(i + 1)) ∈ ∆. If A is deterministic then there is a unique run on every word. A run ρ is accepting if last(ρ) ∈ F . A word w is accepted by A if there is an accepting run of A on w. The language recognized by A is

L(A) = {w ∈ Σ∗| w is accepted by A} .

A language L ⊆ Σ∗ is called regular if there exists an NFA A such that L = L(A). We denote the class of all regular languages by REG.

Remark 2.1. For every L ⊆ Σ∗, the following are equivalent 1. there exists an NFA A such that L = L(A),

2. there exists a DFA A such that L = L(A).

A (nondeterministic) finite ω-automaton (ω-NFA) A = (Q, Σ, Q_in, ∆, Ω) consists of a finite set of states Q with the set of initial states Qin⊆ Q, an

input alphabet Σ, a transition relation ∆ ⊆ Q × Σ × Q and an acceptance condition Ω ⊆ Qω. As in the case of NFA, A is deterministic (ω-DFA) if |Q_in| = 1 and |{q0 _{∈ Q | (q, a, q}0_{) ∈ ∆}| ≤ 1, for all q ∈ Q and all a ∈ Σ.}

In this case, we also use a (partial) function δ : Q × Σ → Q to denote the transition relation ∆ and write qin to denote the set Qin= {qin}.

A run ρ of A on a word w ∈ Σω is an infinite sequence of states ρ = ρ(0)ρ(1) · · · such that ρ(0) ∈ Qin, and for every i ∈ N, we have

(ρ(i), w(i), ρ(i + 1)) ∈ ∆. If A is deterministic then there is a unique run on every word. A run ρ is accepting if ρ ∈ Ω. A word w is accepted by A if there is an accepting run of A on w. The language recognized by A is

L(A) = {w ∈ Σω| w is accepted by A} .

Let col : Q → [n] be a priority function, also called coloring function, which assigns to every state in Q a priority, also called color, from [n], for some n ∈ N. We extend the function col to sequences of states by defining col(ρ) = col(ρ(0))col(ρ(1)) · · · . A parity acceptance condition is defined by

Ωparity= {ρ ∈ Qω | min{Inf(col(ρ))} is even} ,

i.e., a run ρ of an automaton with a parity acceptance condition, denoted by parity-NFA (parity-DFA), is accepting if the minimal priority seen infinitely often in ρ is even.

Let F ⊆ P(Q) be a set of subsets of Q. A Muller acceptance condition is defined by

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i.e., a run ρ of an automaton with a Muller acceptance condition, denoted by Muller-NFA (Muller-DFA), is accepting if the set of states seen infinitely often in ρ is contained in F .

A language L ⊆ Σωis called ω-regular if there exists a parity-NFA A such that L = L(A). The class of all ω-regular languages is denoted by REGω.

Remark 2.2. For every L ⊆ Σω_{, the following are equivalent}

1. there exists a parity-NFA A such that L = L(A), 2. there exists a parity-DFA A such that L = L(A), 3. there exists a Muller-NFA A such that L = L(A), 4. there exists a Muller-DFA A such that L = L(A),

We call the parity and Muller acceptance conditions, which depend on the infinity set of a run, strong acceptance conditions. However, we also consider acceptance conditions which depend on the occurrence set of a run which are called weak. A weak-parity acceptance condition is defined by

Ωweak-parity = {ρ ∈ Qω | min{Occ(col(ρ))} is even} ,

i.e., a run ρ of an automaton with a weak-parity acceptance condition, de-noted by weak-parity-NFA (weak-parity-DFA), is accepting if the minimal priority occurring in ρ is even.

We call a weak-parity acceptance condition E-acceptance condition if col(q) ∈ {0, 1} for every q ∈ Q, and it is called A-acceptance condition if col(q) ∈ {1, 2} for every q ∈ Q. Thus, a run ρ of an automaton with an E-acceptance condition, denoted by E-NFA (E-DFA), is accepting if ρ visits a state with priority 0 at least once, and for an automaton with an A-acceptance condition, denoted by A-NFA (A-DFA), a run is accepting if it never visits a state with priority 1.

In the remainder of this work we will also write col or F for the acceptance condition instead of Ω.

2.2.2 Pushdown Systems, Pushdown Automata

A pushdown system (PDS) S = (Q, Γ, ∆, q_in) consists of a finite set of states Q with the initial state qin ∈ Q, a stack alphabet Γ with an initial stack

symbol ⊥ /∈ Γ, and a transition relation ∆ ⊆ Q × Γ⊥× Q × Γ⊥≤2 ,

where by Γ⊥we denote the set Γ ∪ {⊥}. Moreover, the transition relation ∆

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2.2 Automata

if A = ⊥ then α = γ⊥ with γ ∈ Γ≤1, otherwise α ∈ Γ≤2, i.e., the initial stack symbol ⊥ can neither be written nor deleted from the stack.

A transition δ = (q, A, p, α) ∈ ∆ is called a push-transition if |α| = 2, it is called a skip-transition if |α| = 1, and δ is a pop-transition if α = ε. We say that S is deadlock-free if for every q ∈ Q and every A ∈ Γ⊥ there exist

p ∈ Q and α ∈ Γ_⊥≤2 such that (q, A, p, α) ∈ ∆.

A stack content is a word from Γ∗⊥ where we assume the leftmost symbol to be the top of the stack. A configuration is a pair (q, γ) consisting of a state q ∈ Q and a stack content γ ∈ Γ∗⊥. We define the stack height of a configuration (q, γ) by

sh(q, γ) = |γ| − 1 .

Moreover, we write (q, γ) 7−S (q0, γ0) if there exists (q, γ(0), q0, α) ∈ ∆ and

γ0 = αγ(1) · · · γ(|γ| − 1).

We extend the notion of PDS to pushdown machines, pushdown au-tomata and pushdown transducers. A pushdown machine is a pushdown system augmented by an input alphabet. For pushdown automata, input alphabets and acceptance conditions are attached, and for pushdown trans-ducers, input and output alphabets, and output functions are provided.

A (nondeterministic) pushdown machine (PDM) M = (Q, Σ, Γ, ∆, qin)

consists of a finite set of states Q with the initial state qin ∈ Q, an input

alphabet Σ, a stack alphabet Γ with the initial stack symbol ⊥ /∈ Γ, and a transition relation

∆ ⊆ Q × Γ⊥× Σε× Q × Γ⊥≤2 ,

where Γ⊥ = Γ ∪ {⊥} and Σε = Σ ∪ {ε}. Analogously to PDS, the initial

stack symbol ⊥ marks the bottom of the stack and can neither be written nor deleted from the stack. For a transition δ = (q, A, a, p, α), we call δ a push-, skip- or pop-transition depending on the length of α as for PDS. Furthermore, δ is an ε-transition if a = ε and it is called non-ε-transition otherwise. A PDM M is deterministic (DPDM) if the transition relation ∆ satisfies

{(q0, α) | (q, A, a, q0, α) ∈ ∆}+

{(q0, α) | (q, A, ε, q0, α) ∈ ∆}≤ 1 for all q ∈ Q, all a ∈ Σ, and all A ∈ Γ⊥. In this case we use a (partial)

function δ : Q × Γ⊥× Σε→ Q × Γ⊥≤2 to denote the transition relation ∆.

For two configurations (q, γ), (q0, γ0) ∈ Q × Γ∗⊥, we write (q, γ) 7−aM

(q0, γ0) if there exists (q, γ(0), a, q0, α) ∈ ∆ and γ0= αγ(1) · · · γ(|γ| − 1). A run ρ of M on a finite word w ∈ Σ∗is a finite sequence of configurations ρ = (q0, γ0) · · · (qr, γr) such that

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1. (q0, γ0) = (qin, ⊥), and

2. for every i ∈ [r], there exists ai ∈ Σε such that (qi, γi) ai

7−M(qi+1, γi+1)

and a0· · · ar= w, and

3. {(q, α) | (qr, γr(0), ε, q, α) ∈ ∆} = ∅.

The last item of this definition requires that no execution of an ε-transition is possible from the last configuration of a run of M on a finite word.

A run ρ of M on an infinite word w ∈ Σωis defined as an infinite sequence of configurations ρ = (q0, γ0)(q1, γ1) · · · such that

1. (q0, γ0) = (qin, ⊥), and

2. for every i ∈ N, there exists ai ∈ Σε such that (qi, γi) ai

7−M (qi+1, γi+1)

and a₀a1· · · = w.

Notice, that if M is deterministic then, for every word w ∈ Σ∗∪ Σω_{, if there}

exists a run of M on w then there is exactly one unique run. We say that a PDM M has the continuity property if for every word w ∈ Σω there exists a run of M on w.

A pushdown automaton (PDA) P = (Q, Σ, Γ, ∆, q_in, F ) consists of a PDM MP = (Q, Σ, Γ, ∆, qin) and a set of final states F ⊆ Q. A PDA P is

deterministic (DPDA) if MP is deterministic. We say that a PDA P has the continuity property if MP has the continuity property.

For two configurations c, c0 ∈ Q × Γ∗⊥, we write c 7−aP c0 if c a

7−_MP c0. A

run ρ of P on a word w ∈ Σ∗ is a run of MP on w. A run ρ is accepting if last(ρ) ∈ F . A word w is accepted by P if there is an accepting run of P on w. The language recognized by P is

L(P) = {w ∈ Σ∗ | w is accepted by A} .

A language L ⊆ Σ∗ is called (nondeterministic) contextfree if there exists a PDA P such that L = L(P), and it is called deterministic contextfree if there exists a DPDA P such that L = L(P). We denote the class of all contextfree languages by CFL and the class of all deterministic contextfree languages by DCFL. It is well-known that REG ( DCFL ( CFL [GG66]. Lemma 2.3.

1. For every PDA P, a PDA P0 can be constructed such that L(P) = L(P0) and P0 has the continuity property.

2. For every DPDA P, a DPDA P0 can be constructed such that L(P) = L(P0) and P0 has the continuity property.

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2.2 Automata

The first statement of Lemma 2.3, can easily be established by extend-ing the PDA by just one non-acceptextend-ing sink state which is reached from every configuration via an ε-transition and where every further computa-tion remains forever. The construccomputa-tion for the deterministic case, the second statement of this lemma, involves elimination of loops consisting solely of ε-transitions (cf. [GG66]).

Now, we define pushdown automata on infinite words. An ω-pushdown automaton (ω-PDA) P = (Q, Σ, Γ, ∆, q_in, Ω) consists of a PDM MP = (Q, Σ, Γ, ∆, qin) and an acceptance condition Ω ⊆ (Q × Γ∗⊥)ω. Again, P

is deterministic (ω-DPDA) if MP is deterministic.

A run ρ of P on a word w ∈ Σω is a run of MP on w. A run ρ is accepting if ρ ∈ Ω. A word w is accepted by A if there is an accepting run of A on w. The language recognized by A is

L(A) = {w ∈ Σω| w is accepted by A} .

Let col : Q → [n] be a coloring function, for some n ∈ N. We extend the function col to configurations by defining col(q, γ) = col(q) for every state q ∈ Q and every stack content γ ∈ Γ∗⊥, i.e., the color of a configuration depends only on the state and not on the stack content of the configuration. Now, we consider the acceptance conditions presented in the previous subsection and define the corresponding ω-pushdown automata. A run ρ of a parity-PDA (parity-DPDA) P = (Q, Σ, Γ, ∆, q_in, col) is accepting if

min{Inf(col(ρ))} is even .

A run ρ of a Muller-PDA (Muller-DPDA) P = (Q, Σ, Γ, ∆, q_in, F ) is accept-ing if the set of states seen infinitely often in ρ is contained in F , i.e.,

Inf(Pr0(ρ)) ∈ F .

A language L ⊆ Σω is called (nondeterministic) ω-contextfree if there exists an parity-PDA P such that L = L(P), and it is called deterministic ω-contextfree if there exists an parity-DPDA P such that L = L(P). We denote the class of all ω-contextfree languages by CFLω, and the class of

all deterministic ω-contextfree languages by DCFLω. For these classes the

proper inclusion holds as well, REGω ( DCFLω ( CFLω [CG78].

Remark 2.4. For every L ⊆ Σω, the following are equivalent 1. there exist parity-PDA P such that L = L(P).

2. there exist Muller-PDA P such that L = L(P).

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1. there exist parity-DPDA P such that L = L(P). 2. there exist Muller-DPDA P such that L = L(P). Lemma 2.6.

1. For every parity-PDA P, a parity-PDA P0 can be constructed such that L(P) = L(P0) and P0 has the continuity property.

2. For every parity-DPDA P, a parity-DPDA P0 can be constructed such that L(P) = L(P0) and P0 has the continuity property.

Analogously to Lemma2.3, the construction for parity-PDA utilizes non-determinism and for parity-DPDA the crucial point is again the elimination of loops consisting of ε-transitions (cf. [GG66] and [CG78]). Unless oth-erwise stipulated in the remainder of this work we assume every PDA and every ω-PDA to have the continuity property.

We continue by defining weak ω-pushdown automata. A run ρ of a weak-parity-PDA (weak-parity-DPDA) P = (Q, Σ, Γ, ∆, q_in, col) is accepting if

min{Occ(col(ρ))} is even .

For E-acceptance and A-acceptance conditions we abbreviate the correspond-ing pushdown automata by E-PDA (E-DPDA) and A-PDA (A-DPDA), re-spectively.

We complete by defining pushdown transducers, PDM provided by out-put. A pushdown transducer (PDT) T = (Q, ΣI, ΣO, Γ, ∆, qin, λ) consists

of a PDM MT = (Q, ΣI, Γ, ∆, qin), an output alphabet ΣO and a partial

output function λ : Q → Σ_O. A pushdown transducer T is deterministic (DPDT) if MT is deterministic. A PDT T is a nondeterministic transducer (NFT) if Γ = ∅, i.e., MT can be seen as a finite automaton, since it has no access to the stack. An NFT is deterministic (DFT) if MT is deterministic. We will omit Γ in the description of NFT and DFT.

A run ρ of T on a word w ∈ (Σ_I)∗ is a run of MT on w. A DPDT T defines a partial function fT : (ΣI)∗ → ΣOas follows. For a word w ∈ (ΣI)∗,

let ρ be the unique run of T on w and last(ρ) = (q, γ), for some state q ∈ Q and some stack content γ ∈ Γ∗⊥, then fT(w) = λ(q). We will also

use another definition of DPDT where the output function is of the form λ : Q × Γ⊥ → ΣO. In this case the output symbol not only depends on the

state but also on the top stack symbol of the last configuration of the run of the DPDT T , i.e., for a word w ∈ (Σ_I)∗, and the unique run ρ of T on w with last(ρ) = (q, γ), fT(w) = λ(q, γ(0)). Note, that the definitions are

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2.2 Automata

2.2.3 Trees and Tree Automata

For a set X, an X-tree is a prefix closed set T ⊆ X∗, i.e., for w ∈ X∗ and x ∈ X if wx ∈ T then also w ∈ T . The elements of T are called nodes and the node ε is called root of T . For a node wx ∈ T with w ∈ X∗ and x ∈ X, wx is called child or successor of w, and w is the parent or predecessor of wx. If T = X∗ then it is called full infinite X-tree.

For an alphabet Σ, a Σ-labeled X-tree is a pair (T, t) where T is an X-tree and t : T → Σ is a function assigning to each node from T a symbol from Σ. We call a labeled tree (T, t) full if T is full. By XΣ we denote the

set of all full infinite Σ-labeled X-trees. To simplify our notation we will sometimes write t instead of (X∗, t) for a tree in XΣ.

For a tree t ∈ X_Σ and a set Y we define the Y -widening of t as a Σ-labeled (X × Y )-tree wide_Y(t) = t0 such that t0(z) = t(PrX(z)), for every

node z ∈ (X × Y )∗. Furthermore, we say that a tree t ∈ XΣ is regular if it is

generated by a finite transducer, i.e., if there is a DFT T = (Q, X, Σ, δ, qin, λ)

such that for every node w ∈ X∗, fT(w) = t(w).

For a finite set S, we denote the set of all positive Boolean formulas over propositional variables from S (where the formulas true and false are also allowed) by B+(S). For Boolean formulas we assume, that ∧ has precedence over ∨. Notice that with this definition, every formula from B+(S) is in disjunctive normal form (DNF). We denote such formulas ϕ in DNF also as sets ϕ = {ψ₁, . . . , ψk} of conjuncts where we denote the conjuncts ψi also

as sets ψi ⊆ S of propositional variables. A set S0 ⊆ S satisfies a formula

ϕ ∈ B+(S) if and only if ϕ is true when assigned true to all elements in S0 and false to all elements in S \ S0.

To be able to navigate through a tree we define sets of directions. For a finite set X, let Dir = {↓_x| x ∈ X}, Dir↑ = Dir ∪ {↑}, Dir = Dir ∪ { },

and Dir_↑, _{= Dir ∪ {↑, }. For all w ∈ X}∗ and x ∈ X, we define w·_{= w,} w· ↓x = wx, and wx· ↑ = w.

Let X be a finite set. An alternating parity two-way tree automaton (parity-A2TA) A = (Q, Σ, δ, q_in, col) consists of a finite set of states Q with the initial state q_in, a labeling alphabet Σ, a coloring function col : Q → [n], for some n ∈ N, and a transition function δ : Q × Σ → B+(Dir_↑,× Q).

A run of A on a full infinite Σ-labeled X-tree t ∈ XΣ is a not necessarily

full (Q × X∗_{)-labeled N-tree (T, ρ) such that the following conditions hold.} 1. ε ∈ T and ρ(ε) = (qin, ε).

2. If y ∈ T with ρ(y) = (q, w) and δ(q, t(w)) = ϕ then there is a conjunct ψ = {(d0, q0), . . . , (dk−1, qk−1)} ⊆ Dir↑, × Q in ϕ such that the set of

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A run (T, ρ) is accepting if all its paths satisfy the parity condition col, i.e., for each path π through T (starting in ε) min{Inf(col(ρ(π)))} is even, where ρ(π) is the infinite sequence of labelings of π and col is extended to labelings (q, w) ∈ Q × X∗ such that col(q, w) = col(q) for every q ∈ Q and w ∈ X∗.

A tree t ∈ XΣ is accepted by A if there is an accepting run (T, ρ) of A

on t. The tree language recognized by A is L(A) = {t ∈ XΣ| t is accepted by A} .

We call a parity-A2TA A an alternating parity one-way tree automaton or just alternating parity tree automaton, denoted by A1TA or parity-ATA, if the transition function δ does not use the directions ↑ and _{, i.e.,} it is of the form δ : Q × Σ → B+(Dir × Q). We call a parity-ATA A over Σ-labeled an alternating parity finite automaton, denoted by parity-AFA if |X| = 1, in this case we omit the Dir component in the transition function, i.e., for parity-AFA the transition function is of the form δ : Q × Σ → B+(Q). Let X = {x0, . . . , xk} for some k ∈ N. A parity-ATA is called

nondeter-ministic, denoted by parity-NTA or parity-N1TA, if the transition function is of the following form. For all q ∈ Q and all a ∈ Σ, δ(q, a) has the form

n _ j=0 (↓x0, q j 0) ∧ . . . ∧ (↓xk, q j k) for some n ∈ N.

Notice that for every parity-ATA there is an equivalent parity-NTA, in par-ticular for every parity-AFA there is also an equivalent parity-NFA.

Theorem 2.7 ([MS95]). For every parity-ATA A over Σ-labeled X-trees there is a parity-NTA N over Σ-labeled X-trees such that L(A) = L(N ). Remark 2.8 ([KV99]). For every parity-NTA A over Σ-labeled (X × Y )-trees there is a parity-NTA B over Σ-labeled X-)-trees such that t ∈ L(B) if and only if wideY(t) ∈ L(A).

An alternating (one-way) pushdown tree automaton (APDTA) P = (Q, Σ, Γ, qin, δ, Ω) over Σ-labeled X-trees consists of a finite set Q of states

with the initial state qin, a labeling alphabet Σ, a stack alphabet Γ with the

initial stack symbol ⊥ /∈ Γ, a transition function δ : Q × Σε× Γ⊥→ B+(Dir × Q × Γ⊥≤2) ,

where Γ⊥= Γ∪{⊥} and Σε= Σ∪{ε}, and an acceptance component Ω which

is either a parity acceptance condition col : Q → [n], for some n ∈ N, (in this case the automaton is denoted by parity-APDTA) or a Muller acceptance condition F ⊆ P(Q) (Muller-APDTA). As for pushdown word automata we

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2.2 Automata

assume that the initial stack symbol ⊥ neither can be written to nor be deleted from the stack.

A run of a APDTA P on a Σ-labeled X-tree t ∈ XΣ is a not necessarily

full (X∗× Q × Γ∗_{⊥)-labeled N-tree (T, ρ) such that the following conditions}

hold.

1. ε ∈ T and ρ(ε) = (ε, q_in, ⊥).

2. If y ∈ T with ρ(y) = (w, q, γ) and δ(q, t(w), γ(0)) = ϕ then there is a conjunct ψ = {(d0, q0, α0), . . . , (dk−1, qk−1, αk−1)} ⊆ Dir × Q × Γ⊥≤2

in ϕ such that the set of successors of y in T is precisely {y · i | i ∈ [k]} and ρ(y · i) = (w · d_i, qi, αiγ(1) · · · γ(|γ| − 1)) .

A run (T, ρ) is accepting, if all its paths satisfy the acceptance condition Ω. That means, that for each infinite path π through T (starting in ε), in case of a parity acceptance condition min{Inf(col(ρ(π)))} has to be even, and Inf(Pr1(ρ(π))) ∈ F has to be satisfied in case of a Muller acceptance

condition. Here ρ(π) is the infinite sequence of labelings of π, the coloring function col is extended to labelings (w, q, γ) ∈ X∗ × Q × Γ∗_{⊥ such that}

col(w, q, γ) = col(q) for every w ∈ X∗ and every configuration (q, γ) ∈ Q×Γ∗⊥. Notice that Pr1projects to the set of states Q, i.e., Pr1(w, q, γ) = q

for every w ∈ X∗ and every configuration (q, γ) ∈ Q × Γ∗⊥..

A tree t ∈ X_Σ is accepted by P if there is an accepting run (T, ρ) of P on t. The tree language recognized by P is

L(P) = {t ∈ XΣ| t is accepted by P} .

Let X = {x0, . . . , xk} for some k ∈ N. An APDTA P is called

nondeter-ministic (NPDTA) if, for every q ∈ Q and every A ∈ Γ⊥, either for all a ∈ Σ

there is some (q0, α) ∈ Q × Γ_⊥≤2 such that δ(q, a, A) = (_{, q}0, α) or, for all a ∈ Σ, δ(q, a, A) has the form

n _ j=0 (↓x0, q j 0, α j 0) ∧ . . . ∧ (↓xk, q j k, α j k)

for some n ∈ N. We denote an NPDTA with a parity (Muller) acceptance condition by NPDTA (Muller-NPDTA). Notice, that like for parity-NTA, the nonemptiness problem for parity-NPDTA, which is to decide, given a parity-NPDTA P over Σ-labeled X-trees whether L(P) 6= ∅, is shown to be decidable.

Theorem 2.9 ([KPV02]). The nonemptiness problem for parity-NPDTA is decidable.

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2.3 Infinite Games

2.3.1 Gale-Stewart Games

We consider so called Gale-Stewart games, turn-based infinite two-player games of perfect information, which were introduced in [GS53]. We present here a slightly modified version of the original definition.

Let Σ_I and Σ_O be two alphabets and let Σ = Σ_I × Σ_O. We call Σ_I input alphabet and Σ_O output alphabet. An ω-language L ⊆ Σω _{defines the}

Gale-Stewart game Γ(L). The game Γ(L) is played by two players, Player I (the input player which will also be denoted by Player 1) and Player O (the output player, also denoted by Player 0) in rounds i ∈ N. The two players pick letters from their respective alphabets in alternation. In every round i, first Player I picks a letter ai from ΣI and then Player O (being aware of

the choice ai of Player I) picks a letter bi from ΣO.

A play of Γ(L) is a sequence a₀, b0, a1, b1, a2, b2, . . . of letters which yields

two ω-words, the input word α = a₀a1a2· · · constructed by Player I and the

output word β = b0b1b2· · · produced by Player O. The language L is used

to determine the winner of the play, i.e., it provides the winning condition. Player O wins the play if and only if the ω-word α_β = α(0)_β(0) α(1)_β(1) α(2)_β(2) · · · induced by the play is contained in L.

A strategy for Player I is a functionσI: (ΣO)∗→ ΣI, and a strategy for

Player O is a function σO: (ΣI)∗ → ΣO. Consider a play a0, b0, a1, b1, . . .

and its induced ω-word α_β. The play is consistent with a strategy σI if

α(n) =σI(prefn(β)), for all n ∈ N. The play is consistent with a strategyσO

if β(n) =σO(prefn+1(α)), for all n ∈ N. A strategyσis winning for Player i,

for i ∈ {0, 1}, if every play which is consistent with σ is won by Player i.

We say that Player i wins Γ(L) if there is a winning strategy for Player i. A game Γ(L) is determined if it is won by either of the two players.

Consider Σω _{as a topological space. Open sets are languages of the form}

W ·Σωwhere W ⊆ Σ∗and closed sets are complements of open sets. The class of open sets is denoted byΣ1 and the class of closed sets is denoted byΠ1.

A language L ⊆ Σω is a Borel set if it is obtained from open and closed sets by repeatedly applying countable unions and countable intersections. Based on the classesΣ1 andΠ1 the finite levels of the Borel Hierarchy are defined

inductively, Σn+1 is the class of countable unions of Πn-sets, and Πn+1 is

the class of countable intersections ofΣn-sets, for n > 0. Let B(Σn) denote

the boolean closure of the n-th level of the Borel Hierarchy. The full Borel hierarchy is obtained by including the classes Σα, for countable ordinals α.

Figure2.1illustrates how the classes are related.

Theorem 2.10 ([Mar75]). Γ(L) is determined, for every Borel set L ⊆ Σω. We will also represent strategies by functions which take into account the

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2.3 Infinite Games Σ1 B(Σ1) Π1 Σ2 B(Σ2) Π2 Σ3 Π3 · · · ( ( ( ( ( ( ( ( ( (

Figure 2.1: Finite Borel hierarchy

whole history of a play and not only sequences produced by the opponent, i.e., of the form σI: (ΣI× ΣO)∗ → ΣI and σO: (ΣI× ΣO)∗ΣI → ΣO. Let

a0, b0, a1, b1, . . . be a play and α_β its induced ω-word. The play is

con-sistent with σI if α(n) = σI(prefn(α_β)), for all n ∈ N. It is consistent

with σO if β(n) = σO(prefn(α_β)α(n)), for all n ∈ N. Notice, that both

representations can be converted into each other.

For a class L of ω-languages, we refer to a Gale-Stewart game Γ(L) as an L-game if L ∈ L.

2.3.2 Games on Graphs

A game graph G = (V, V0, V1, E, vin) consists of a (possibly countably

infi-nite) directed graph (V, E) with set V of vertices and set E ⊆ V ×V of edges, a partition V₀∪ V₁ of the set of vertices V and the initial vertex v_in ∈ V . A vertex v ∈ V is called reachable if there is a path from v_in to v. We say that a game graph G is deadlock-free if for every vertex v ∈ V there is a vertex v0 ∈ V such that (v, v0) ∈ E, i.e., every vertex has at least one outgoing edge. For the reason of convenience, in the following, we assume deadlock-free game graphs.

A game G = (G, Ω) consists of a game graph G and a winning condi-tion Ω ⊆ Vω. A play in G is built up by moving a token on the game graph G. Initially, the token is placed on vin. If the vertex v where the

token is currently located is in V_i, then Player i has to choose an outgoing edge (v, v0) ∈ E and the token is moved to the vertex v0. Thus, a play in G is an infinite sequence of vertices ρ ∈ Vω such that ρ(0) = vin and

(ρ(n), ρ(n + 1)) ∈ E, for all n ∈ N. The winning condition Ω consists of all plays winning for Player O. A play ρ is winning for Player I if ρ ∈ Vω\ Ω.

Let col : V → [n] be a coloring function assigning to every vertex in V a color from [n], for some n ∈ N. Analogously to acceptance conditions for automata (cf. 2.2), we call a winning condition Ω parity winning condition if Ω = {ρ ∈ Vω | min{Inf(col(ρ))} is even}, it is called weak-parity winning condition if Ω = {ρ ∈ Vω | min{Occ(col(ρ))} is even}. Moreover, reach-ability winning conditions corresponds to E-acceptance and safety winning conditions to A-acceptance conditions. We denote a parity (weak-parity,

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reachability or safety) game by G = (G, col). Furthermore, for F ⊆ P(V ), A Muller winning condition is defined by Ω = {ρ ∈ Vω | Inf(ρ) ∈ F } and a Muller game is denoted by G = (G, F ).

A strategy for Player i is a function σ: V∗Vi → V such that for every

w ∈ V∗Vi, we have (last(w),σ(w)) ∈ E. A strategy σ is called positional

if σ(w) = σ(w0) holds for all w, w0 ∈ V∗Vi with last(w) = last(w0), i.e., the

choice of the next move does not depend on the whole play prefix but only on the current vertex. A play ρ is consistent with a strategy σ for Player i

if ρ(n + 1) =σ(pref_n+1(ρ)) for every n ∈ N with ρ(n) ∈ Vi. A strategy σ

is winning for Player i if every play ρ which is consistent withσ is winning

for Player i. We say that Player i wins a game G if there exists a winning strategy for Player i in G. A game G is determined if it is won by either of the two players.

Theorem 2.11 ([EJ91], [Mos91]). Parity games are determined with posi-tional winning strategies.

We will also use the following representations for strategies. Notice, that every play prefix can be described by a sequence of edges instead of a sequence of vertices. Hence, a strategy for Player i can be defined by

σ: E∗ → V such that for every η ∈ E+ with last(η) = (v, v0), for some v ∈ V

and v0 ∈ V_i, we have (v0,σ(η)) ∈ E, and if vin ∈ Vi, then (vin,σ(ε)) ∈ E.

Moreover, a strategy for Player i can also be defined by σ: E∗ → E such

that for every η ∈ E+ with last(η) = (v, v0), for some v ∈ V and v0 ∈ Vi, we

haveσ(η) = (v0, v00), for some v00 ∈ V , and if vin ∈ Vi, then σ(ε) = (vin, v00),

for some v00∈ V . Finally, one can define a strategy for Player i by a function

σ: V∗Vi → E such that for every w ∈ V∗Vi, we have σ(w) = (last(w), v),

for some v ∈ V . Notice that all these representations can be converted into each other.

Pushdown Game Graphs

Let S = (Q, Γ, ∆, qin) be a PDS. The induced pushdown graph G(S) =

(V, E), also called configuration graph, is an infinite directed graph where the set of vertices V = {(q, γ) | q ∈ Q, γ ∈ Γ∗⊥} is the set of configurations, and for two configurations v, v0 ∈ V , there exists the edge (v, v0_{) ∈ E if and}

only if v 7−S v0. Notice that G(S) is deadlock-free if S is deadlock-free.

Now, a partition Q0∪ Q1 of the set of states Q induces the game graph

G = (V, V0, V1, E, vin) which is called pushdown game graph, where (V, E) =

G(S), the partition V0 ∪ V1 of the set of configurations V is defined by

Vi = {(q, γ) ∈ V | q ∈ Qi}, for i ∈ {0, 1}, and vin= (qin, ⊥).

A pushdown game G = (G, Ω) consists of a pushdown game graph G = (V, V0, V1, E, vin) and a winning condition Ω ⊆ Vω. In the most cases we

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2.3 Infinite Games

will consider winning conditions which do not depend on the stack contents but only on the states of configurations, i.e., winning conditions Ω of the following form. For every ρ ∈ Vω, if ρ ∈ Ω then every ρ0 ∈ Vω _with

Pr0(ρ) = Pr0(ρ0) is also contained in Ω. In such cases, we will also write

Pr0(Ω) for the winning condition instead of Ω. Pushdown games with more

general winning conditions (which also depend on the stack contents) are studied e.g. in [CDT02], [Ser04] and [Fin05]. In Chapter3, we will introduce a winning condition which also depends on the stack contents, more precisely on the stack heights.

For a PDS S = (Q, Γ, ∆, q_in) and its induced pushdown game graph G = (V, V0, V1, E, vin), let col : Q → [n] be a coloring function extended to

configurations via col(v) = col(Pr0(v)), for every configuration v ∈ V . We

refer to a pushdown game G = (G, col) with a parity (weak-parity, reachabil-ity, or safety) winning condition col as a parity (weak-parreachabil-ity, reachabilreachabil-ity, or safety) pushdown game. Moreover, we refer to a pushdown game G = (G, F ) with a Muller winning condition F ⊆ P(Q) as a Muller pushdown game.

2.3.3 Game Reduction, Game Simulation

Let G = (G, Ω) and G0 = (G0, Ω0) be games on graphs G = (V, V0, V1, E, vin)

and G0= (V0, V₀0, V₁0, E0, v_in0 ) with winning conditions Ω and Ω0, respectively. We say that G = (G, Ω) is reducible to G0 = (G0, Ω0) if there exists a memory structure M = (Q, δ, qin) consisting of a finite set Q of states with

initial state q_in and an update function δ : Q × V → Q such that 1. V0 = V × Q, V₀0 = V0× Q, V10 = V1× Q and v0_in= (vin, qin)

2. ((v, q), (v, q)) ∈ E0 if and only if (v, v) ∈ E and δ(q, v) = q

3. Player i wins a play ρ in G if and only if Player i wins the play ρ0 = (v0, q0)(v1, q1) · · · in G0 induced by ρ such that (v0, q0) = v0in

and (vn, qn) = (ρ(n), δ(ρ(n), qn−1), for all n > 0.

Game reduction is used to transfer games with complex winning con-ditions which are hard to handle to games with possibly simpler winning conditions. The drawback which has to be accepted is the growth of the size of the game graphs which get larger by taking the cartesian product with the memory structure. A prominent example is the reduction of Muller games to parity games using the latest appearance record [Tho95].

Now, we formulate a more general notion of game simulation. We say that G = (G, Ω) is simulated by G0 = (G0, Ω0) if there exist two functions f : V∗Vi → (V0)∗Vi0 and g : E0× V → V such that

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2. if σ0: (V0)∗V_i0 → E0 is a winning strategy for Player i in G0 then σ: V∗Vi → V with σ(w) = g(σ0(f (w)), last(w)), for every play

pre-fix w ∈ V∗Vi, is a winning strategy for Player i in G.

The function f transfers a play prefix w ∈ V∗Vi in G into a play prefix

w0 ∈ (V0)∗V_i0 in G0. To deduce a winning strategy σ in G from a winning

strategyσ0 in G0 the function g is used which maps an edge in E0 (the next

move given byσ0) and the current vertex from Vi to the vertex which has to

be chosen next.

In contrast to game reduction where the size of the game graph increases in the reduction process, the more general game simulation allows to transfer games on large (infinite) game graphs to games on smaller (finite) game graphs. A game simulation transferring parity pushdown games (on infinite pushdown graphs) to parity games on finite game graphs [Wal96,Wal01] will be used in Section6.

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Chapter 3

Pushdown Games and

Pushdown Winning Strategies

From the Büchi-Landweber result and the known correspondence between finite automata and monadic second-order logic it follows that Gale-Stewart games with specifications definable in monadic second-order logic can be solved with winning strategies which are again definable in monadic second-order logic. This fact raises the question concerning the conceptual con-nection between formats of game specifications and corresponding solutions. This problem can be viewed as the following reformulation of Church’s Prob-lem focusing on the formats of the game specifications and their solutions. Given a winning condition L in a specific format, i.e., L is an ω-language from a particular class recognizable by a certain type of automata or de-finable in some specific logic, it is to decide whether there exist a winning strategyσfor the winner such thatσcan be implemented in the same format

as the specification L, i.e., σ is definable in the same logic or realizable by

a transducer of the same type. This relation between formats of game spec-ifications and corresponding solutions has been analyzed for several regular classes. Selivanov established a tight connection for the class of aperiodic regular languages [Sel07]. He showed that games with aperiodic regular win-ning conditions are determined with winwin-ning strategies realizable by aperi-odic transducers. Rabinovich and Thomas established analogous result for a number of sublogics of the monadic second-order logic [RT07], among them first-order logic over (N, <), the extension of first-order logic over (N, <) by modular counting quantifiers, first order logic over (N, S) with successor re-lation S and the quantifier-free first-order logic over (N, 0, +1) called strictly bounded logic. Moreover, examples of logics where winning strategies of the same format don’t suffice were exhibited. Chaturvedi et al. studied sub-classes of star-free regular languages in [COT11] where piecewise testable languages and languages from the dot-depth hierarchy are considered.

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In this chapter, we extend these results to contextfree specifications, hence, we address the relation between the formats of winning conditions and corresponding winning strategies for contextfree Gale-Stewart games as well as parity games played over pushdown game graphs. We consider sev-eral classes of contextfree specifications which we introduce in Section 3.1. We establish a tight connection between the formats of specifications and their solutions for a number of cases. Furthermore, we also present some cases where this correspondence fails. To prove this results, which we state in Section3.2, we first recall the technique of Kupferman and Vardi [KV00a] in Subsection3.2.1. The proofs are presented in Subsection3.2.2.

3.1 Classes of Contextfree Languages

In this section we define several classes of contextfree ∗-languages and con-textfree ω-languages. Various number of such classes is conceivable which can be defined by different kinds of pushdown automata recognizing those classes. We distinguish the types of pushdown automata by several prop-erties of the underlying pushdown machine on the one hand and by the underlying acceptance conditions on the other hand.

For pushdown automata on finite words we consider acceptance by fi-nal states, and for those on infinite words we consider parity acceptance, as defined in Section 2.2.2. Beyond that, we introduce a new kind of ac-ceptance conditions for ω-pushdown automata, so-called stair acac-ceptance conditions [LMS04]. Notice that this notion is also naturally carried over to winning conditions for games on pushdown game graphs, called stair winning conditions.

Intuitively, for a finite or infinite path through a configuration graph, a configuration is said to be a stair configuration if no subsequent configuration of smaller stack height exists in this path.

Definition 3.1 (Stairs). Let V = Q × Γ∗⊥ be a set of configurations over some set Q of states and a pushdown alphabet Γ. Define the functions StairPositions : V+∪ Vω _{→ 2}N _{and Stairs : V}+_{∪ V}ω_{→ V}+_{∪ V}ω _by

StairPositions(w) = {n ∈ N | ∀m ≥ n : sh(w(m)) ≥ sh(w(n))},

and Stairs(w) = w(n0)w(n1) · · · , where n0 < n1 < · · · is the ascending

enumeration of StairPositions(w), for w ∈ V+∪ Vω_.

Using this notion we now define stair acceptance conditions for push-down automata (and stair winning conditions for games on pushpush-down game graphs). Let M = (Q, Σ, Γ, ∆, q_in) be a PDM. A stair parity acceptance condition is given by a coloring function col : Q → [n], for some n ∈ N. A

A study of pushdown games